For convenience I've attached both the question and my attempt at the solution. Please see figures attached. From the quadratic equation for the desired phase margin, I selected the w = 2.682 solution because I thought that we couldn't have negative values for w. (i.e. not physical) However the correct solution comes about when w = +1.491, but the solution to my equation indicates w = -1.491. How do we know when a negative frequency indicates an invalid solution, or a valid one where we can simply neglect the negative sign?
I believe it's a simple sign error in your equations: You have which then becomes then then and so on ....